Setting of the magnetic structure of chiral kagome antiferromagnets by a seeded spin-orbit torque

The current-induced spin-orbit torque switching of ferromagnets has had huge impact in spintronics. However, short spin-diffusion lengths limit the thickness of switchable ferromagnetic layers, thereby limiting their thermal stability. Here, we report a previously unobserved seeded spin-orbit torque (SSOT) by which current can set the magnetic states of even thick layers of the chiral kagome antiferromagnet Mn3Sn. The mechanism involves setting the orientation of the antiferromagnetic domains in a thin region at the interface with spin currents arising from an adjacent heavy metal while also heating the layer above its magnetic ordering temperature. This interface region seeds the resulting spin texture of the entire layer as it cools down and, thereby, overcomes the thickness limitation of conventional spin-orbit torques. SSOT switching in Mn3Sn can be extended beyond chiral antiferromagnets to diverse magnetic systems and provides a path toward the development of highly efficient, high-speed, and thermally stable spintronic devices.


Electrical switching measurement
The electrical switching experiments were performed in a PPMS (Quantum Design, Inc.) and probe station (Lakeshore cryotronics) system. Two distinct switching protocols were used in these investigations. In protocol I (Fig. S1 A), first a write pulse of variable pulse length [switching scheme 1, 2 and 3 in Fig. S2] was used. After a specific delay period (0.5 s), the resultant magnetic orientations were probed by measuring the transverse voltage in the Hall bar with a 1 mA d.c. read current. In switching protocol II (Fig. S1 B), a single current pulse is

I. Structural analysis and magnetic properties of Mn3Sn thin film
The texture of the polycrystalline Mn3Sn film was studied by X-ray diffraction using a Bruker D8 Discover four circle diffractometer operated with monochromatized Cu-K-1 radiation and a six-circle diffractometer using focused Ga-K- radiation from a Ga-jet x-ray source. Fig. S3 a shows the 2- scan on a logarithmic intensity scale in the range between 2 = 20° and 80° collected with the Bruker D8 Discover using a two-dimensional pixel detector operated in one-dimensional scan mode. The measurement was carried out on a 100 nm thick Mn3Sn thin film. Within this angular interval, all reflections could be observed with exception of the (0002) reflection near 2 = 40° which is too weak, while in other studies it has been detected (29,47). Already on a qualitative basis this suggests that our film is highly textured with a preferential orientation of the (112 ̅ 0) crystal face parallel to the film surface. Moreover, in our case the (112 ̅ 0) reflection is about three times as strong as compared to the (202 ̅ 1) reflection. The latter should be the strongest one in the ideal case of a polycrystal with a completely random orientation of the crystallites, which is observed in the XRD patterns of Refs (29,47). The inset shows the reciprocal space map (RSM) in the vicinity of the (202 ̅ 1) reflection. In the RSM data which were collected using a six-circle diffractometer and a Ga-jet X-ray source under grazing incidence (incidence angle µ = 1°) of the primary beam, three diffraction rings are observed which are identified as reflections belonging to TaN (101 ̅ 1) and to the Mn3Sn (0002) and (202 ̅ 1) reflection. The intensity ratio between the latter is equal to about 1:5. In order to evaluate the presence of a texture in the deposited film we have quantitatively analyzed the reflection intensities by integrating the peaks and deriving an effective multiplicity as shortly outlined as follows: The integrated intensity [I(hkil)] of a polycrystalline (powder) sample at the Bragg angle  is given by (49) where |F(hkil)|, Vx, vuc and µ represent the structure factor magnitude, the volume of the crystal, the volume of the unit cell (uc) and the linear absorption coefficient, respectively. Importantly, the factor H represents the multiplicity (i.e. the statistical weight) of the crystal faces in a polycrystalline sample. For instance, in space group (SGR) P63/mmc the multiplicity of the basis-pinacoid {0001} is equal to 2, while for the hexagonal prism {101 ̅ 0} it is equal to 6.

II. Model tight-binding calculations
The purpose of this section is to show that the net magnetic moment of a kagome triangle allows to determine the shape of the resistivity tensor, even though it is not the order parameter. For the purpose of generality, we do not restrict our calculations to Mn3Sn but model a generic kagome magnet that exhibits a phase with a negative vector spin chirality and is AB stacked. Since Mn3Sn d bands dominate near the Fermi level, we consider 5 d orbitals for each of the three lattice sites giving us 30 bands in total. The electronic Hamiltonian reads The first term comprises the hopping terms, accounting for the electrons' kinetic energy.
An electron with orbital and spin is annihilated at site (operator † ) and is created with orbital at site with the same spin (operator ). This process is quantified by the hopping amplitude . Note, that not all hopping amplitudes are independent of each other.
, where , and are the Pauli matrices. The SOC term is an onsite term that mixes different orbitals according to . The third term is the interaction of the electron spin ( vector of Pauli matrices) with the magnetic texture ( unit vectors). This is also an onsite term but it mixes different spin directions. The strength of this coupling is = 0.2 eV.
We diagonalize the Hamiltonian to determine the band structure and the eigenvectors | ⟩, both dependent on the reciprocal vector k. We calculate the Berry curvature tensor for band n From the Berry curvature, we can calculate the intrinsic contribution to the Hall conductivity as the integral over all occupied states (energy below the Fermi energy ) First, we simulate the six different equilibrium magnetic phases ( = 30°, 90°, 150°, 210°, 270°, 330°) that we have discussed in the paper. Our numerical results (three exemplary phases in Fig. S6) reveal that the conductivity tensor can be decomposed as This means, that the plane of Hall transport is perpendicular to the net magnetization, similar to the situation in a ferromagnet. However, the mechanism behind it is very different. The net moment is way too small to explain the measured signal. The origin is a set of broken symmetries by the arrangement of magnetic moments. This effect remains, even if we artificially fix the moments at angles 120° with respect to each other, so that the net magnetization is exactly zero. As is shown in Figure S7, this only changes the Hall conductivity very slightly. This means, the net moment is not the order parameter but under the experimental conditions, it still allows to determine the shape of the resistivity tensor. In the experiment, we measure the xy element which is why we observe the maximum signal for the field-switching mechanism that involves switching between the = 90° and = 270° configurations where the sine has its extrema. For the other four configurations, that are relevant for the SOT switching, the sine is only ±0.5. In (C) the Hall conductivity is shown for both configurations. The difference is only marginal.

III. Theoretical simulation of current induced switching
We have carried out theoretical simulation of current induced switching as discussed in the previous publication (29). When a current is applied along x, the SHE in W injects a spin current with a polarization s ∥ ± y into the Mn3Sn layer, i.e., along = 0° or = 180°, depending on the sign of the current. If the current density is sufficiently large, the injected spin current orients all the magnetic configurations along these two unstable configurations (cf. energy maxima in Fig. 1C in the main text). Using atomistic simulations this change in the magnetic state can be calculated, as illustrated in Fig. S8 (A, B). The absence of the bias field (Hx) leads to a perfect alignment of m along the two directions characterized by = 0°, 180°, once the applied current is larger than the critical current. These configurations correspond to two energy maxima shown in Fig. 1C. For this reason, once the current is turned off, domains in the configuration = 0° will relax either to = 30° or = 330° (two equally close energy minima). The probability for both relaxation processes is equal and since the two configurations are characterized by mz = +0.5m and mz = -0.5m, respectively, the measured xy Hall resistivity cancels. The same argument holds for the domains that are in a = 180° state.
In the presence of Hx, the metastable states never reach exactly the injected spin orientations = 0° and = 180° but approach close to these values depending on the current density. The direction (sign with respect to z) of H × s determines whether is slightly smaller or larger than these saturation values. When the current is turned off, the system then relaxes to the nearest energy minimum depending on the sign of H × s. This means that by reversing the current direction (and therefore s), the magnetic configurations are switched from either = 330° to 30° or from = 210° to 150° and vice-versa (depending on the Hx direction), thereby changing the sign of the Hall voltage. For both metastable states, this corresponds to a switching between configurations with a projected net moment of mz = ± 0.5 |m|. Therefore, the theoretically highest possible Hall signal is half as large as that in the fieldswitching experiment, i.e., ~ ± 25 mΩ for the present study.

II
Current induced switching experiment is shown in Fig. S9 A using Protocol II in the absence of bias field (Hx=0). The non-linear behavior of Rxy reflects that a finite Rxx contribution to Rxy due to a small mismatch in the voltage contacts and this helps to calibrate the device temperature. Taking the Rxy value at Jx = 0 for zero field at different measurement temperature (inset of Fig. S9 A), the device temperature is found to be 435 K when switching occurs. The dependence of Jc on the pulse length and as a function of temperature, is summarized in Fig. S9 B (from Protocol II).  Fig. S10 A shows the zero-field longitudinal resistivity ( ) as a function of temperature over the temperature range from 5K to 400K for the 30 nm Mn3Sn thin film. It exhibits a metallic transport behavior as expected with residual resistivity ratio (RRR) ~ 1.75 which demonstrates a high quality of thin film.

V. Magneto-transport properties of Mn3Sn thin films
is 180±10  cm at room temperature for the different thicknesses of Mn3Sn. We have also measured the longitudinal resistance (Rxx) of the device during the writing pulse application. Lower inset of Fig. S10 A shows that the resistance of the device is increasing more than 12  for a write pulse of magnitude ±9V. By comparing this change in Rxx with temperature dependent Rxx measurements (upper inset of Fig.   S10 A) we conclude that the temperature of the device is beyond 400 K for a voltage close to ±9V. Variable temperature Hall measurements were carried out to investigate the change in Hall signal when the temperature of the system is increased. A few representative ′ vs Hz

VI. Thermal randomization in Mn3Sn in absence of spin current
The anomalous Hall and current induced switching of single layer 40 nm Mn3Sn is shown in Fig. S11 A and B. Although there is no spin current source, ′ rapidly decreases at a critical current. This clearly show that there is no role of spin orbit torque for the setting of magnetic states or chiral spin rotation (30) rather the magnetic states sets by thermally.

VII. Current-induced switching of different thicknesses of Mn3Sn/W bi-layers
Here

VIII. Numerical modeling of the Joule heating and cooling during current pulse application
Here we present the details of the finite element method for the Joule heating and cooling simulations that were used to estimate the variation of the device temperature during the current-induced switching. The 'Joule heating' and the 'heat transfer' modules of the COMSOL multi-physics software were used for this study. First, a rectangular writing pulse (with amplitude V0 and pulse length, L) were applied on the Hall bar device (Fig. S16). The time (t) dependent evolution of the temperature (T) during and after application of the writing pulse was estimated by numerically solving the transient heat-diffusion equation: where is the density, is the specific heat capacity, T is the temperature,  is the thermal heat conductivity, J is the applied current density and is electrical conductivity of the device. Since all our measurements were carried out in vacuum condition, convective cooling processes were not included in our simulation. The Hall bar of our experiment is comprised of a stack sequence Al2O3 (substrate) / Mn3Sn (30 nm) / W (8 nm) / TaN (3 nm). Here, in COMSOL, this Hall bar is modeled as a single layer metallic film whose effective electrical conductivity ( ) and thermal conductivity () were estimated from the and  of Mn3Sn and W. Since these devices consist mainly of the Mn3Sn layer, other materials properties were approximated for Mn3Sn only.  (1) Black points (+Hx and -Jx) in Fig. 5, A and C: Estimated when Hx is positive and the current sweep is from zero to the negative direction as shown here in Fig. S17, A. The value of ∆ ′ is positive in this case.
(2) Red points (+Hx and +Jx) in Fig. 5, B and C: Estimated when Hx is positive and the current sweep is from zero to the positive direction as shown here in Fig. S17, B. The value of ∆ ′ is negative in this case.
(3) Purple points (-Hx and -Jx) in Fig. 5, B and C: Estimated when Hx is negative and the current sweep is from zero to the negative direction as shown in here in Fig. S17, C. The value of ∆ ′ is negative in this case.
(4) Blue points (-Hx and +Jx) in Fig. 5, B and C: Estimated when Hx is negative and the current sweep is from zero to the positive direction as shown in here in Fig. S17, D.
The value of ∆ ′ is positive in this case.

X. Switching mechanism comparison with standard FM switching in a CoFeB film
Here we compare the switching mechanism of Mn3Sn with the standard switching mechanism of a perpendicular magnetic anisotropic (PMA) ferromagnet Ta (5 nm) / CoFeB (1 nm) / MgO (2 nm) /Ta (3 nm). Results of current-induced switching (protocol-I) at different Hx for the PMA ferromagnet are summarized in Fig. S18. In the case of CoFeB, Jc shows a strong dependence on Hx (Fig. S18 B). It decreases monotonically with increasing Hx. However, in case of Mn3Sn, Jc does not show any dependence on the applied Hx (Fig. 5B, main text). This is because the field only serves as a bias and becomes only relevant when the critical current is reached and the system is in a switchable state. Also, in the case of the normal PMA sample ∆ ′ is almost independent of the applied Hx (Fig. S18 C). This is because the switching happens always between two distinct magnetic states. Intriguingly, ∆ ′ in Mn3Sn exhibits a strong non-monotonic dependence on the applied magnetic field as is evident from Fig. 5C in the main text.

XI. Combination of field and current induced switching
In the main manuscript, we have discussed two distinct switching mechanisms: A field switching mechanism, when the field Hz is oriented along =± 90°, and a SSOT switching, for which a field perpendicular to the kagome plane (Hx) provides a bias ∆ during the switching. Next, we explore the combination of both effects. We apply a field of ±100 mT at an angle with respect to x and within the xz plane (Fig. S19 A). A very interesting finding is that there are now 4 distinct ′ states whose magnitudes are highly sensitive to as shown in Fig. S19 B. At = 0°, ′ = ±20 m is measured; the same value as discussed earlier for H oriented along ± . Once H is tilted away from x, we observe that there are two branches for ′ corresponding to ±H, shown in Fig. S19 B as blue and red. For each along each branch, the sample can be switched hysteretically by current, between two distinct ′ states, whose difference decreases with increasing . The average values of these states are shifted either upwards (blue) and downwards (red) from zero, as is varied away from zero. ′ can also be switched from the red to the blue hysteresis curve when the field is reversed but only if the critical current is exceeded. In particular, for = 90°, the maximum values of ′ in each of the branches approaches those values obtained in pure magnetic field switching i.e. ±50 m (see Fig. S10 B). However, here a field of just 100 mT is required as compared to 1T needed for pure field switching. The difference is caused by the heating provided by the current. Note that for = 90° the SOT provided by the current has no effect on the switching and, therefore, cannot provide a seeding layer. Still, it heats up the sample and the magnetic field (Hz along = ±90°) provides the bias throughout the whole Mn3Sn layer. In short, the field-switching mechanism is also strongly affected by current-induced heating in a favorable way. Finally, Fig. S19 C shows how the 4 states can be accessed for = 10° by applying current pulses in the presence of H = ± 0.1 T. We conclude that there are 4 distinct ′ states that can be accessed by a current-induced switching mechanism that is strongly influenced by relatively small magnetic fields. These 4 states do not correspond to four individual magnetic phases but to different ratios of the six energetically preferred states presented in Fig. 1C in the main manuscript.

XII. Fast Magnetization reversal with nanosecond current pulse in presence of a bias
SSOT mediated switching can only take place if the device relaxes from the ordering temperature in presence of SOT provided by a critical spin current density ( * ). Switching experiments using nanosecond current pulses (with fall time ~ 750 ps) lead to very small switching efficiency () since the spin current source turnoff abruptly even before the system relaxes below * . Therefore to overcome this situation, switching experiments were carried out using nanosecond current pulses ranging from 100 ns to 10 ns in presence of a bias (Fig.   S20). It is to be noted that the role of the bias is to provide critical spin current density when the system ( * ) when the system cools below * . We note that the magnitude of the bias voltage (2V) is below the critical current density necessary to switch the device.